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The market events of 2007-2009 have reinvigorated the search for realisticreturn models that capture greater likelihoods of extreme movements. In thispaper we model the medium-term log-return dynamics in a market with bothfundamental and technical traders. This is based on a Poisson trade arrivalmodel with variable size orders. With simplifications we are led to a hybridSDE mixing both arithmetic and geometric Brownian motions, whose solution isgiven by a class of integrals of exponentials of one Brownian motion againstanother, in forms considered by Yor and collaborators. The reduction of thehybrid SDE to a single Brownian motion leads to an SDE of the form consideredby Nagahara, which is a type of "Pearson diffusion", or equivalently ahyperbolic OU SDE. Various dynamics and equilibria are possible depending onthe balance of trades. Under mean-reverting circumstances we arrive naturallyat an equilibrium fat-tailed return distribution with a Student or Pearson TypeIV form. Under less restrictive assumptions richer dynamics are possible,including bimodal structures. The phenomenon of variance explosion isidentified that gives rise to much larger price movements that might have apriori been expected, so that "$25\sigma$" events are significantly moreprobable. We exhibit simple example solutions of the Fokker-Planck equationthat shows how such variance explosion can hide beneath a standard Gaussianfacade. These are elementary members of an extended class of distributions witha rich and varied structure, capable of describing a wide range of marketbehaviours. Several approaches to the density function are possible, and anexample of the computation of a hyperbolic VaR is given. The model alsosuggests generalizations of the Bougerol identity.

A model of returns for the post-credit-crunch reality: Hybrid Brownian motion with price feedback